The rounding and congruence functions include seven basic functions. They all deal with the separation of integer or fractional parts from real and complex numbers: the floor function (entire part function) , the nearest integer function (round) , the ceiling function (least integer) , the integer part , the fractional part , the modulo function (congruence) , and the integer part of the quotient (quotient or integer division) .

The floor function (entire function) can be considered as the basic function of this group. The other six functions can be uniquely defined through the floor function.

Floor

For real , the floor function is the greatest integer less than or equal to .

For arbitrary complex , the function can be described (or defined) by the following formulas:

Examples: , , , , , ,.

Round

For real , the rounding function is the integer closest to (if ).

For arbitrary , the round function can be described (or defined) by the following formulas:

Examples: , , , , , , , .

Ceiling

For real , the ceiling function is the smallest integer greater than or equal to .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , , .

Integer part

For real , the function integer part is the integer part of .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , ,.

Fractional part

For real , the function fractional part is the fractional part of .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , ,.

Mod

For complex and , the mod function is the remainder of the division of by . The sign of for real , is always the same as the sign of .

The mod function can be described (or defined) by the following formula:

The functional property makes the behavior of similar to the behavior of .

Examples: , , , , , .

Quotient

For complex and , the integer part of the quotient (quotient) function is the integer quotient of and .

The quotient function can be described (or defined) by the following formula: